We prove, using variational methods, the existence in dimension two of positive vector ground states solutions for Bose-Einstein type systems. The nonlinear interaction between two Bose fluids is assumed to be of critical exponential type in the sense of J. Moser. For “small" solutions the system is asymptotically equivalent to the corresponding one with power-like nonlinearities, recently deeply studied in higher dimensions. (In collaboration with H. Tavares and J. Zhang, to appear in JDE).